As we have mentioned before, the study of the classical stability of the solutions constructed above is
a very difficult problem. For gravitating solutions coupled to matter fields (and, in particular, with non-Abelian fields)
one must deal with a big set of coupled diferential equations. In our case, an extra complexity is that each solution has different symmetries and,
therefore, the perturbation to be considered must be (in principle) different for each case. For example, in
the case of the black string, the “most dangerous” perturbation (that is, the perturbation that respects the
symmetries of the system and can generate unstable modes) must include tensor, vector and scalar modes, in addition
to the corresponding perturbations to the three degrees of freedom of the NLSM matter field, which makes
the system very difficult to decouple. In fact, the study of the stability of extended objects without matter fields is itself a
complex problem, as can be seen in the pioneer work of Gregory and Laflamme [1]}.
Even more, the generalization to the \(SU(N)\)  case add different factors to deal with in the linearized equations for each group.
Something similar happens with the black hole solutions.
